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Sunday, July 22, 2007

Many Worlds: A brief guide for the perplexed

I added this to the earlier post 50 years of Many Worlds and thought I would make it into a stand alone post as well.

Many Worlds: A brief guide for the perplexed

In quantum mechanics, states can exist in superpositions, such as (for an electron spin)

(state) = (up) + (down)

When a measurement on this state is performed, the Copenhagen interpretation says that the state (wavefunction) "collapses" to one of the two possible outcomes:

(up) or (down),

with some probability for each outcome depending on the initial state (e.g., 1/2 and 1/2 of measuring up and down). One fundamental difference between quantum and classical mechanics is that even if we have specified the state above as precisely as is allowed by nature, we are still left with only a probabilistic prediction for what will happen next. In classical physics knowing the state (e.g., position and velocity of a particle) allows perfect future prediction.

There is no satisfactory understanding of how or exactly when the Copenhagen wavefunction "collapse" proceeds. Indeed, collapse introduces confusing issues like consciousness: what, exactly, constitutes an "observer", capable of causing the collapse?

Everett suggested we simply remove wavefunction collapse from the theory. Then the state evolves in time always according to the Schrodinger equation. In fact, the whole universe can be described by a "universal wave function" which evolves according to the Schrodinger equation and never undergoes Copenhagen collapse.

Suppose we follow our electron state through a device which measures its spin. For example: by deflecting the electron using a magnetic field and recording the spin-dependent path of the deflected electron using a detector which amplifies the result. The result is recorded in some macroscopic way: e.g., a red or green bulb lights up depending on whether deflection was up or down. The whole process is described by the Schrodinger equation, with the final state being

(state) = (up) (device recorded up) + (down) (device recorded down)

Here "device" could, but does not necessarily, refer to the human or robot brain which saw the detector bulb flash. What matters is that the device is macroscopic and has a large (e.g., Avogadro's number) number of degrees of freedom. In that case, as noted by Everett, the two sub-states of the world (or device) after the measurement are effectively orthogonal (have zero overlap). In other words, the quantum state describing a huge number of emitted red photons and zero emitted green photons is orthogonal to the complementary state.

If a robot or human brain is watching the experiment, it perceives a unique outcome just as predicted by Copenhagen. That is, any macroscopic information processing device ends up in one of the possible macroscopic states (red light vs green light flash). The amplitude for those macroscopically different states to interfere is exponentially small, hence they can be treated thereafter as completely independent "branches" of the wavefunction.

Success! The experimental outcome is predicted by a simpler (sans collapse) version of the theory. The tricky part: there are now necessarily parts of the final state (wavefunction) describing both the up and down outcomes (I saw red vs I saw green). These are the many worlds of the Everett interpretation.

Personally, I prefer to call it No Collapse instead of Many Worlds -- why not emphasize the advantageous rather than the confusing part of the interpretation?

Some eminent physicists who (as far as I can tell) believe(d) in MW: Feynman, Gell-Mann, Hawking, Steve Weinberg, Bryce DeWitt, David Deutsch, Sidney Coleman ... In fact, I was told that Feynman and Gell-Mann each claim(ed) to have independently invented MW, without any knowledge of Everett!

3 comments:

Anonymous
said...

Steve,

Thanks for the explanation! I realize now it is more subtle than I thought (I agree the "many worlds" adjective does not help).

My follow-up questions are:

1. This means that the macroscopic observer is described quantum mechanically as well? The Hamiltonian is complicated (persumably the rest of the universe), but the details are irrelevant as long as it is "large".

Then my head hurts thinking about questions like "what are the possible states of the universe", etc... I suppose this should be viewed as an "effective description"---we will never be able to describe everything in gory detail, but the details do not matter so much in most problems of interest.

2. The measurements are "perfect", i.e., it is not possible to have

(up)(device measured down) or(down)(device measured up)

That is, how do we know that what we "measure" is really the actual state (e.g., "apparatus" malfunction)?

PS: Did you get a complimentary Chumby? I gotta have one of those! I hope it does well---I like the concept (especially the completely hackable; amazed it took so long for someone create a cool Open Source gizmo) and I like the guy.

1) Yes, it's hard to think about the qm state of the whole system, but completeness requires that even the measuring device be so describable. As I mentioned in the earlier post, theorists who have to think about the universe as a whole (quantum cosmologists) or about an isolated quantum computer are used to this, and they tend to believe in many worlds. (Otherwise, are "collapses" happening inside your quantum computer? What if particular subroutines inside are very complicated "robots" -- do they collapse the wavefunction in the middle of the calculation? How do I know I am not such a robot?)

2) I assumed a well-behaved measurement. Of course you could have much more complicated superpositions resulting from less idealized interactions of the original spin. Sidney Coleman once said to me "quantum mechanics is just a theory of correlations". I should add him to the list of MW believers, now that I think of it.

I don't have a Chumby -- my son would probably rip it to shreds before I had a chance to hack it. They are pretty cool, but I wonder how big the market is for that kind of thing... (the vc in me has to ask :-)